Properties

Degree 2
Conductor 229
Sign $-0.462 - 0.886i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.23i·2-s + 3-s − 3.00·4-s + 3·5-s + 2.23i·6-s − 2.23i·8-s − 2·9-s + 6.70i·10-s + 3·11-s − 3.00·12-s + 3·15-s − 0.999·16-s − 3·17-s − 4.47i·18-s − 19-s − 9.00·20-s + ⋯
L(s)  = 1  + 1.58i·2-s + 0.577·3-s − 1.50·4-s + 1.34·5-s + 0.912i·6-s − 0.790i·8-s − 0.666·9-s + 2.12i·10-s + 0.904·11-s − 0.866·12-s + 0.774·15-s − 0.249·16-s − 0.727·17-s − 1.05i·18-s − 0.229·19-s − 2.01·20-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(229\)
\( \varepsilon \)  =  $-0.462 - 0.886i$
motivic weight  =  \(1\)
character  :  $\chi_{229} (228, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 229,\ (\ :1/2),\ -0.462 - 0.886i)$
$L(1)$  $\approx$  $0.827711 + 1.36545i$
$L(\frac12)$  $\approx$  $0.827711 + 1.36545i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 229$, \(F_p\) is a polynomial of degree 2. If $p = 229$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad229 \( 1 + (-7 - 13.4i)T \)
good2 \( 1 - 2.23iT - 2T^{2} \)
3 \( 1 - T + 3T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + 4.47iT - 23T^{2} \)
29 \( 1 + 4.47iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 4.47iT - 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 - 8.94iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 8.94iT - 59T^{2} \)
61 \( 1 - 5T + 61T^{2} \)
67 \( 1 - 13.4iT - 67T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 + 13.4iT - 73T^{2} \)
79 \( 1 + 13.4iT - 79T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + 17.8iT - 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.14580004509351118819641553580, −11.61430493196621781706902305185, −10.20704899738732125371168016271, −8.998106452521941953152706182178, −8.768961470526460026351817099329, −7.41739122861685143280509277723, −6.29205951881109080234144756635, −5.78640405170751959347105161981, −4.38492488245376512643072179509, −2.37938663941908880657930341002, 1.65730355152551961546014863016, 2.66907075111205383917085810778, 3.88261708380527167211809567495, 5.42927634164617286025699613855, 6.71778066430661336888582716183, 8.587851934459013834761684572790, 9.288995754727251390007447684528, 9.933228883600645345117312716392, 10.99769158155824883235202256723, 11.75485079284358130848965550474

Graph of the $Z$-function along the critical line